Asymptotics of noncolliding q-exchangeable random walks Article Swipe

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Leonid Petrov , Mikhail Tikhonov · YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2303.02380

We consider a process of noncolliding $q$-exchangeable random walks on $\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk is called $q$-exchangeable if under an elementary transposition of the neighboring steps (down, straight) $\to$ (straight, down) the probability of the trajectory is multiplied by a parameter $q\in(0,1)$. Our process of $m$ noncolliding $q$-exchangeable random walks is obtained from the independent $q$-exchangeable walks via the Doob's $h$-transform for a certain nonnegative eigenfunction $h$ with the eigenvalue less than $1$. The system of $m$ walks evolves in the presence of an absorbing wall at $0$. We show that the trajectory of the noncolliding $q$-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of $q$-distributed random lozenge tilings of sawtooth polygons. In the limit as $m\to \infty$, $q=e^{-γ/m}$ with $γ>0$ fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

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preprint

Language

en

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http://arxiv.org/abs/2303.02380

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https://arxiv.org/pdf/2303.02380

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green

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10

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https://openalex.org/W4323556903

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https://openalex.org/W4323556903

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DOI

https://doi.org/10.48550/arxiv.2303.02380

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Title

Asymptotics of noncolliding q-exchangeable random walks

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preprint

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en

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2023

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2023-03-04

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Leonid Petrov, Mikhail Tikhonov

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https://arxiv.org/abs/2303.02380

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https://arxiv.org/pdf/2303.02380

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green

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https://arxiv.org/pdf/2303.02380

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Concepts

Random walk, Mathematics, Kernel (algebra), Combinatorics, Scaling limit, Ergodic theory, Limit (mathematics), Mathematical analysis, Scaling, Statistics, Geometry

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0

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10

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